The time-reversal and reciprocal properties of the lossless linear wave equation can be utilized to achieve useful effects even in wave-chaotic systems [E. Ott, Chaos in Dynamical Systems (Cambridge University Press, New York, 1993); H.-J. Stöckmann, Quantum Chaos: An introduction (Cambridge University Press, New York, 1999); Y. V. Fyodorov and D. V. Savin, in The Oxford Handbook of Random Matrix Theory, edited by G. Akemann, J. Baik, and P. Di Francesco (Oxford University Press, New York, 2011)] typically endowed with complex boundaries and inhomogeneities [M. Fink, Geophysics 71, SI151 (2006); M. Fink, Phys. Today 50, 34 (1997); M. Fink, et al., Rep. Prog. Phys. 63, 1933 (2000); G. Lerosey, et al., Science 315, 1120 (2007); F. Lemoult, et al., Phys. Rev. Lett. 104, 203901 (2010); F. Lemoult, et al., Phys. Rev. Lett 107, 064301 (2011); G. Lerosey, et al., Phys. Rev. Lett. 92, 193904 (2004); and C. Draeger et al., Phys. Rev. Lett. 79, 407 (1997)]. Wave equations without dissipation are invariant under time reversal. Given any time-forward solution considered as a superposition of traveling waves, there exists a corresponding time-reversed solution in which the individual superposed traveling waves propagate backwards retracing the trajectories of the timeforward solution.
This allows the construction of a time-reversal mirror. In an imaginary ideal situation, one transmits a wave form of finite duration from a localized source in the presence of perfectly reflecting objects and then receives the resulting reverberating wave forms (referred to as the sona) on an array of ideal receivers completely enclosing the region where the source and reflecting objects are located.
After the reverberations “die out”, one then transmits (in the opposite direction) the time-reversed sona signals from the array of receivers. This newly transmitted set of signals essentially undoes the time-forward propagation, producing waves which converge on the original localized source, reconstructing a time-reversed version of the original signal at the localized source. Although real situations deviate from the above described ideal, time reversal in this manner has been effectively realized in acoustics [M. Fink, Geophysics 71, SI151 (2006); M. Fink, Phys. Today 50, 34 (1997); M. Fink, et al., Rep. Prog. Phys. 63, 1933 (2000); G. Lerosey, et al., Science 315, 1120 (2007); F. Lemoult, et al., Phys. Rev. Lett. 104, 203901 (2010); F. Lemoult, et al., Phys. Rev. Lett 107, 064301 (2011); G. Lerosey, et al., Phys. Rev. Lett. 92, 193904 (2004); C. Draeger, et al., Phys. Rev. Lett. 79, 407 (1997); B. T. Taddese, et al., Acta Phys. Pol. A 116, 729 (2009); B. T. Taddese, et al., App. Phys. Let. 95, 114103 (2009); and B. T. Taddese, et al., J. Appl. Phys. 108, 114911 (2010)] and electromagnetic waves [F. Lemoult, et al., Phys. Rev. Lett. 104, 203901 (2010); G. Lerosey, et al., Phys. Rev. Lett. 92, 193904 (2004); S. M. Anlage, et al., Acta Phys. Pol. A 112, 569 (2007); and B. T. Taddese, et al., Electron. Lett. 47, 1165 (2011)], and applications such as lithotripsy [M. Fink, Geophysics 71, SI151 (2006); and M. Fink, et al., Rep. Prog. Phys. 63, 1933 (2000)], underwater communication [M. Fink, Geophysics 71, SI151 (2006); A. Parvulescu, J. Acoust. Soc. Am. 98, 943 (1995); and R. Jackson, et al., J. Acoust. Soc. Am. 89, 171 (1991)], sensing small perturbations [B. T. Taddese, et al., Acta Phys. Pol. A 116, 729 (2009); B. T. Taddese, et al., App. Phys. Let. 95, 114103 (2009); and B. T. Taddese, et al., J. Appl. Phys. 108, 114911 (2010)], and achieving subwavelength imaging [F. Lemoult, et al., Phys. Rev. Lett. 104, 203901 (2010); F. Lemoult, et al., Phys. Rev. Lett 107, 064301 (2011); G. Lerosey, et al., Phys. Rev. Lett. 92, 193904 (2004); and P. Blomgren, et al., J. Acoust. Soc. Am. 111, 230 (2002)] have been developed.
Ideally, for a perfect time-reversal mirror, a large number of receivers are required to collect the sona signals, and the receivers need to cover a surface completely surrounding the source and any reflecting objects (which reflect without loss).
A significant simplification is to enclose the system in a closed, ray-chaotic environment with highly reflecting boundaries. For wavelengths smaller than the enclosure size, propagating waves will (over a sufficiently long duration) reach every point in the environment, allowing a single wave-absorbing receiver to record a single time-reversible sona signal over a long duration of time [C. Draeger, et al., Phys. Rev. Lett. 79, 407 (1997); B. T. Taddese, et al., Acta Phys. Pol. A 116, 729 (2009)].
It has been found that in the presence of boundary reflection loss, a high-quality version of the basic time-reversal reconstruction still occurs at the source, and reception of only a small fraction of the transmitted energy is sufficient for reconstruction of the initial wave form at the source. Nevertheless, such a time-reversal mirror still requires an active source to generate the sona signal. In some cases, it would be better if this step could be eliminated, further simplifying the time-reversal mirror.
Recent studies have investigated the addition of discrete elements with complex nonlinear dynamics to otherwise linear wave-chaotic systems [S.D. Cohen, et al., Phys. Rev. Lett. 107, 254103 (2011); and M. Frazier, et al., Phys. Rev. Lett. 110, 063902 (2013)]. When a discrete nonlinear element is added to the system, excitations at new distinct frequencies are generated from the interaction of the initial wave form with the nonlinear element. This appears as a radiated signal originating at the location of the nonlinear element (which in principle may be unknown). The new wave form propagates through a linear medium and, when time-reversed and retransmitted, will reconstruct the excitations generated at the nonlinear element. This form of nonlinear time reversal differs from wave propagation through a distributed nonlinear medium, in which the time-reversal invariance breaks when shock waves form [M. Tanter, et al., Phys. Rev. E. 64, 016602 (2001); and A. P. Brysev, et al., J. Acoust. Soc. Am. 118, 3733 (2005)].
Time reversal in systems with localized nonlinearities has been demonstrated in several systems, including acoustic waves through materials with defects [T. J. Ulrich, et al., J. Acoust. Soc. Am. 119, 1514 (2006); A. S. Gliozzi, et al., ibid. 120, 2506 (2006); T. J. Ulrich, et al., Phys. Rev. Lett. 98, 104301 (2007); and T. Goursolle, et al., Int. J. Non-Linear Mech. 43, 170 (2008)], as a means of nondestructive evaluation [M. Scalerandi, et al., J. Phys. D: Appl. Phys. 41, 215404 (2008); P. Y. Le Bas, et al., J. Acoust. Soc. Am. 130, EL258 (2011); and F. Ciampa et al., ibid. 131, 4316 (2012)], phase conjugation of light harmonically generated from a nanoparticle [C.-L. Hsieh, et al., Opt. Express 18, 12283 (2010); and X. Yang, et al., ibid. 20, 2500 (2012)], and phase conjugation of acoustically modulated light using a focused ultrasonic signal as a “guide star” for the time-reversed focusing [X. Xu, et al., Nature Photonics 5, 154 (2011)].
In [M. Frazier, et al., Phys. Rev. Lett. 110, 063902 (2013)], nonlinear time reversal was performed using microwaves incident upon a harmonically driven diode, generating intermodulation products. A drawback of the technique is the need to use an active nonlinearity (a driven diode to create intermodulation products) instead of a completely passive element. Also missing is a quantitative model to describe and understand the nonlinear time-reversal physics. Furthermore, the technique was applied to develop a method of exclusive communication, which appears to be rate limited by the length of the sonas necessary to transmit information.
It is therefore desirable to create a wave chaotic system with a passive discrete nonlinear element in a nonlinear time-reversal mirror, and to construct a model system to simulate propagation through the nonlinear wave-chaotic system in question.